A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness collection. This talk will discuss a toolkit for the numerical manipulation of multiprojective varieties that operates on witness collections.

Coming from Algebraic Statistics, our main motivating example is the likelihood correspondence. The likelihood correspondence is a biprojective variety encapsulating the geometry of maximum likelihood estimation (MLE), up to a model's Zariski closure. We will use the toolkit to understand the likelihood geometry of a model. Time permitting, we will see how the Huh-Sturmfels involution is used to derive a stopping criterion for a monodromy method that determines the maximum likelihood degree of submodels.